Weighted 1MP and MP1 inverses for operators

Abstract

The main aim of this paper is to extend the concepts of the 1MP and MP1 inverses defined for rectangular complex matrices. We present the weighted 1MP and MP1 inverses for a bounded linear operator between two Hilbert spaces as two new kinds of generalized inverses. The notions of the weighted 1MP and MP1 inverses are new in the context of rectangular complex matrices too. We establish a number of characterizations and some representations of the weighted 1MP and MP1 inverses. Several operator equations are solved applying the weighted 1MP and MP1 inverses. A special case of one of these equations is the normal equation which is related with the least-squares solution. As consequences of our results, we obtain new properties of the 1MP and MP1 inverses.